Most dealerships won't let you, or would only let you charge $500-$1000 of the purchase price.
It's because the transaction fees are a percentage of the sale, not a flat fee, and it would cut into the profit margin. (Places like Best Buy account for people using plastic when they set their prices, but car dealerships don't, because they negotiate each sale individually.) Since they're not allowed to charge you an extra fee for using the card (as per the merchant agreement with Visa/Mastercard/whomever), they just say tell you no.